Abstract
In this article, we study the distribution of index of Farey fractions by providing asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with \(\mathcal {B}\)-free denominators. Additionally, we reconsider the square-free case earlier done in Alkan et al. (Ramanujan J 16(2):131–161, 2008), and obtain new results for moments of indices with square-free denominators. We also obtain higher level correlation measures of the index function, generalizing earlier known results on two level correlations.
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Acknowledgements
The authors would like to thank Tomos Parry for valuable inputs during the preparation of this article. The first author acknowledges support from the Science and Engineering Research Board, Department of Science and Technology, Government of India, under grant SB/S2/RJN-053/2018. The authors are grateful to the referee for valuable suggestions in an earlier version of the article.
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Chahal, B., Chaubey, S. & Goel, S. On the distribution of index of Farey sequences. Res. number theory 10, 27 (2024). https://doi.org/10.1007/s40993-024-00511-y
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DOI: https://doi.org/10.1007/s40993-024-00511-y